Intersections of commutants with closures of derivation ranges
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- by Domingo A. Herrero PDF
- Proc. Amer. Math. Soc. 74 (1979), 29-34 Request permission
Abstract:
The norm closure of the set ${\mathcal {A}_w}(\mathcal {X}) = \cup \;\{ {\text {Ran}}{({\delta _A})^{ - w}} \cap \{ A\} ’:A \in \mathcal {L}(\mathcal {X})\}$, where ${\delta _A}$ denotes the inner derivation induced by the operator A, ${\text {Ran}}{({\delta _A})^{ - w}}$ is the weak closure of the range of ${\delta _A}$ and $\{ A\} ’$ is the commutant of A, is disjoint from the open dense subset $\mathcal {B}(\mathcal {X}) = \{ T \in \mathcal {L}(\mathcal {X})$: T has a nonzero normal eigenvalue} for every complex Banach space $\mathcal {X}$. For a Hilbert space $\mathcal {H}$, $\mathcal {L}(\mathcal {H}) = \mathcal {B}(\mathcal {H}) \cup {\mathcal {A}_w}{(\mathcal {H})^ - }$, where the bar denotes norm closure.References
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Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 74 (1979), 29-34
- MSC: Primary 47B47
- DOI: https://doi.org/10.1090/S0002-9939-1979-0521868-3
- MathSciNet review: 521868